Square
|vertex_count = 4 |edges_length = 4l |surface_area = l^2 |vertices = 4 points |edges = 4 line segments |faces = 1 square |image1 = Square.png |symmetry = Square dihedral symmetry (D4)}}A square is the 2 dimensional hypercube. It has the schläfli symbol \{4\} , as it is a four-sided polygon. Other names of square are called tetragon or tetrasquaron (Using Googleaarex's polytope naming system). Its Bowers acronym is also "square". Under the elemental naming scheme it is called a geogon,' aerogon', or staurogon. Squares are one of the three regular polygons that tile the plane. The others are the equilateral triangle and regular hexagon. The tiling is called a square tiling, and has four squares around each vertex. The reason why squares can tile the plane is that the interior angle of a square is (1/n) * 360 degrees, where n is a whole number. If n is not a whole number, then you cannot tile the plane. The symmetry group of a square is D4, since there are four possible reflections that will leave the square unchanged: through the two lines joining the midpoints of opposite edges, and through the two lines joining the opposite vertices of the square. Four squares can fit between a vertex, at least in Euclidian geometry. Hypercube Product The square can be expressed as a product of hypercubes in two different ways: * \{\}^2 (line prism) * \{4\} (square) Symbols A square can be given several Dynkin symbols and their extensions, including: *x4o (fully regular) *x x (rectangle) *qo oq&#zx (rhombus) *xx&#x (trapezoid) *oqo&#xt (kite) *oooo&#xr (generic quadrilateral) Structure and Sections Sections The square can be thought of as infinitely many line segments stacked on each other in the y direction, or a prism with a line segment as the base. As such, when viewed from a side, the sections are identical lines. It is composed of two pairs of parallel line segments. When viewed from a vertex, the point will expand into a line of length \sqrt{2} before turning back to a point. Hypervolumes * vertex count = 4 * edge length = 4l * surface area = l^2 Subfacets * 4 points (0D) * 4 line segments (1D) * 1 square (2D) Radii * Vertex radius: \frac{\sqrt{2}}{2}l * Edge radius: \frac{1}{2}l Angles *Vertex angle: 90º Equations All points on the surface of a square with side length 2 can be given by the equation \max(x^2,y^2) = 1 A square rotated by 45º, with side \sqrt{2} , can be given by the equation |x|+|y| = 1 Vertex coordinates The vertex coordinates of a square of side 2 are (±1, ±1). The dual orientatoin of this square, with side length \sqrt{2} has coordinates: *(±1,0) *(0,±1) Notations *Toratopic notation: || *Tapertopic notation: 11 Related shapes *Dual: Self dual * Vertex figure: Line segment, length \sqrt{2} (also diagonal) Coordinate System The coordinate system associated with the square is plane cartesian coordinates. This coordinate system has a length element with length \text{d}s^2 = \text{d}x^2 + \text{d}y^2 and an area element \text{d}A = \text{d}x \text{d}y . See Also Category:Shape Category:2 dimensional Category:Polygons Category:Regular polygons Category:Uniform polygons Category:Hypercubes Category:Rotatopes Category:Toratopes Category:Tapertopes Category:Quadrilateral